Time-changes of self-similar Markov processes

A NNALES DE L ’I. H. P., SECTION B

J. V ^UOLLE -A ^PIALA

Time-changes of self-similar Markov processes

Annales de l’I. H. P., section B, tome 25, n

^o

4 (1989), p. 581-587

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Time-changes of self-similar Markov processes

J. VUOLLE-APIALA

Aarhus University, Department ^of Mathematics, Ny Munkegade, ^{DK-8000 Aarhus} C., ^Denmark Ann. Inst. Henri Poincaré,

Vol. 25, n° 4, 1989, p. 581-587. Probabilités et Statistiques

ABSTRACT. - Let

Xt

^{be a}

03B2-self-similar, 03B2

^>

0,

transient Markov _process

on

(0, oo).

^We show that if

XTt ^(Tt

^{is the}

^right

continuous inverse of a

continuous additive functional

At)

^{is an} 03B1-self-similar Markov _process,

a > 0,

^then

A result

concerning time-changes

^{of a transient}

Levy

process is also

given.

Key ^{words :} Self-similar, Markov process, time-change, Levy process.

RESUME. - Soit

Xt

^un processus

p-self-similaire

^{transient et de Markov}

sur

(0, oo), ~3 > o.

^Notons

Tt

l’inverse continu a droite du fonctionnelle additive

At.

^{Nous montrons} que si

XTt

^{est un processus} a-self-similaire et de

Markov, Cl>O,

^alors

Un resultat concernant le

changement

^de temps d’un processus de

Levy

transient est

également

^donne.

582 J. VUOLLE-APIALA

0. INTRODUCTION

a-self-similar Markov _processes

(a-ssmp)

^on

(0, oo)

^were introduced

by

J.

Lamperti [5].

^The process

(Xt, P")

^{with a state} space

(0, oo)

^{is called}

a-ssmp,

ex>O,

if there exists a Borel

semigroup (Pi, ))t _> o

^on

(0, oo ) (0, oo ) satisfying

(i) Po ( ~ ) = I~

(ii) Pt (x, aa A)

^{for all}

t > 0, a > 0, x E (0, oo ), A E ~ (0, oo ),

such that

(Xt, Px)

^{is a time}

homogeneous

^{Markov process with a transition}

function

(Pt ( ,

^{and with}

sample paths

^{that are P~-almost}

surely right-continuous

^with left limits for all

x E (0, oo).

It was

proved

ⁱⁿ

[6]

^that every a-ssmp with "nice

paths" (see Notation)

on

(0, oo )

^{has a} weak dual with

respect

^{to the measure} dx. In this note we

apply

this result and

characterize,

^the

theory developed

ⁱⁿ

[3]

^as

a main

tool,

^{all the}

possible

ways ^to

time-change

^{a transient} self-similar process

(in ^fact,

^an a-ssmp is transient iff it is

cotransient;

^see

Proposition)

to another self-similar process. We also obtain ^a result

concerning

^time-

changes

^{of a}

Levy

process. For

simplicity,

^{we assume}

a > 0,

but the results

can

easily

^be

generalized

^to

negative

^a.

1. - NOTATION. DEFINITIONS

Q notes the space of all functions ^o from

[0,

^~

(0, oo) U ~ ~ ~ ^(A

denotes the

point

^{used as a}

"graveyard";

^we assume A is an isolated

point),

^which

satisfy

(a)

^for

^t>_0; ~(t)=0 ~;

(b) co

^is

right

continuous and co or has left limits on

[0, oo)

^at every Such Q is called the _space of "nice

paths".

DEFINITION. - Let a > 0 be

given.

A stochastic process

(Xt,

^{with a}

state space

(0,

^{is called} a-ssmp ^on

(0, oo )

^{if the}

following

^is

satisfied: there exists a Borel

semigroup (Pt ( , ))t >_ o

^on

^{(0, oo)}

^{X ~}

^{(0, oo)}

with the

properties:

(i) 1’0 ( ~ ) = I~

(ii) Pt (x,

^{for all}

t ~ 0, a > 0, x E (0, oo ), A E ~ (0, oo ),

such that

(Xt, P~)

^{is a time}

homogeneous

Markov process with ^a transition function

(Pt ( , )t > o

^{and for}

^{x E (0, ao ).}

Remark I. -

^{It was}

proved

ⁱⁿ

[4]

that every a-ssmp ^o.n

(0, oo ) automati- cally

^is

strongly

^Markov.

A Markov _process

(Xt, ^Px, x E (0, oo))

^{is said to} be in weak

duality

^with

a Markov process

(Xt, Px, x E (0, oo))

^with

respect

^{to a a-finite measure}

Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques

583

SELF-SIMILAR MARKOV PROCESSES

r~, if for all

bounded f, g

ⁱⁿ

~ (o, oo)

Let

(Xt, P~)

be in weak

duality

^with

(Xt, Px)

^with

respect

^{to a measure}

(Xt, PX)

^{is said to be}

transient,

^if

for all _x, all

bounded, non-negative

Borel functions

f

^on

(0, oo)

^with

compact support (see

alternative definitions for transience in

[2]).

^{If the}

dual process

(Xt, Px)

^is

transient,

^then

P~)

^{is said to be} cotransient.

Remark 2. - It was shown in

[6]

^{that an} a-ssmp ^on

(0, oo )

^{has a weak}

dual,

^with

respect

^{to the measure}

xl~«- ~ dx,

and the dual _process is also

an a-ssmp.

2. - THEOREMS

We assume

throughout

this paper that

P~)

is transient

(as

^{we shall}

see in

Proposition,

for self-similar processes the transience is

equivalent

to the

cotransience). According

^to

[6], (XTt, ^P~)

^{is an a-ssmp if}

(Xt, P")

^is

p-ssmp

^and

Tt

^{is the}

right

continuous inverse of an additive functional k We shall show that this is the

only possible

way ^to time-

change (Xt, Px)

^{to an} a-ssmp.

PROPOSITION. - Let

(Xt, Px)

^{be a}

03B2-ssmp

^on

(0, r), [3 >

^{0. Then it is}

transient

iff

it is cotrasient.

Proof. - According

^to

[4] (Th. 2.3)

^and

[5] (Th. 4.1),

^{there is one to}

one

correspondence

^{between a}

p-ssmp Xt

^on

(0, oo)

^{and a}

Levy

process

Zt

^on

( - oo, + oo) (that is, Zt

^{is a}

strong

^Markov process which have

stationary independent

increments and

right

continuous

paths

^{with left}

limits)

^defined

by

^where

Tr

^{is the}

right

continuous inverse of

an additive functional

It is also

easily

^{seen that}

Xt

is transient iff

Zr

is transient. Now for

Zt

there exists ^a weak dual

2~

^with

respect

^{to the}

Lebesgue

^{measure such}

that also

Zt

^{is a}

Levy

process. As shown in

[6], starting

^from

Zt

^{one can}

construct a

03B2-ssmp t,

^{which is a weak dual to}

Xt

^with respect ^{to the}

584 J. VUOLLE-APIALA

measure dx. Now

Xt

is transient iff

Zt

is transient and so it suffices

to show that

Zt

is transient iff it is cotransient. If

(Zt, QZ)

^{is a}

Levy

process then

Zt

^under

QZ

^{has the same} distribution as

z + Zt

^under

Q°

and thus _easy calculations show that

Zt

^under

QZ

^{has the same} distribution

as

z - Zt

^under

Q°.

This shows that

Zt

is transient iff

Zt

is transient and

gives

thus the assertion.

In the

proof

^{of the}

following

^theorem

actually

cotransience

(and

^not

transience)

^{is used.}

THEOREM 2.1. - Let

(Xt, Px)

^{be a transient}

03B2-ssmp

^on

(0, oo), (3

^>

0,

^and

let

At be a

continuous

additive functional

^with

Tt

^{as the}

right

^continuous

inverse,

^{i. e.}

If

^{the process}

Px)

^is 03B1-ssmp, then there ^exists k > 0 such that

Proof. -

As mentioned in Remark

2, (Xt, P")

^{has a weak dual}

(Xt, P")

with

respect

^{to the measure}

x l ~~ -1 dx

such that also

(Xt, P")

^is

p-ssmp.

Let

At

^{be a} continuous additive functional of

Xt

^{and let}

(XTt, P"), Tt

^is

the

right

continuous inverse of

At,

^{be an} a-ssmp. Let further vA be the Revuz measure

corresponding

^to

At. According

^to the result of Getoor and

Sharpe [3]

for _any

bounded, non-negative

Borel function

/

^with

compact support.

The

right

^{side of}

(2.1)

^is

equal

^to

~0f(XTt)dt }y1/03B2-1 dy, ^which,

because of the

a-self-similarity

^of

Px),

^is

equal

^to

Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques

SELF-SIMILAR MARKOV PROCESSES 585

From

(2.1)

^and

(2 . 2)

^we

obtain, by substituting z = a°‘ y,

The

03B2-self-similarity

^of

(Xt, Px) implies

This, together

^with

(2. 3), gives

Because

Xt

is cotransient we have

LJ, f ’ (x)

^{+ oo , V x. Thus}

Applying

^the well-known

uniqueness

result for a Haar measure we obtain

which

gives

Remark. - The

special

^case of Theorem 1 is oc ⁼

P,

^which

gives At

^{= kt.}

This means that the

only possible

^{way to}

time-change

^{a transient}

P-ssmp

to another

P-ssmp

^{is a linear}

time-change.

In

[5]

^J.

Lamperti

introduced a continuous additive functional

where

(Xt, P~)

^{is a}

p-ssmp

^on

(0, oo).

^{He showed that if}

Tt

^{is the}

right

continuous inverse of

At,

^{then the}

time-changed

process

(XTt, ^P~)

^{is a}

strong ^Markov process ^on

(0, oo)

such that it is

multiplicatively invariant,

I. e.

A) = Qt (ax, a A),

^{for all}

Xe(0, oo ), oo ),

where

Qt ( , )

^{is a} transition function for

(XTt, ^Px).

The

following

^theorem says that

this, possibly multiplied by

^a constant, is the

only

way ^to

time-change (X~, P")

^{to a}

multiplicatively

^invariant

586 J. VUOLLE-APIALA

process:

THEOREM 2.2. - Let

Px)

^{be a transient}

03B2-ssmp

^on

(0, oo)

^{and let}

At

be a continuous additive

functional of Px)

^and

Tt

^the

right

^continuous

inverse

of Ar. If Px)

^is

multiplicatively invariant,

then there exists k > 0 such that

The

proof

^{is similar to that} of Theorem 2.1 and will therefore be omitted.

Finally,

^{we shall}

present

^{a result}

concerning time-changes

^{of a}

Levy

process. It ^was

already

^remarked

by Lamperti [5]

^that

(Yt)

^{is a}

multiplica- tively

^invariant

strong

^Markov process with "nice

paths" (see Notation)

on

(0, oo)

^iff

(Zt) _ (log Yt)

^{is a}

Levy

process ^on

( -

oo, -

oo).

^{We will}

show

THEOREM 2.3. - The

only possible

way ^to

time-change

^{a transient}

Levy

process ^on

( -

~, ⁺

oo )

^{to another}

Levy

process is ^a linear

time-change t - kt,

^k>o.

We need the

following

^Lemma:

LEMMA. - Let

(Yr, P~)

^{be a}

transient, multiplicatively

^invariant strong Markov _process with "nice

paths" (see Notation)

^on

(0, oo).

^{Then the}

only possible

^{way to}

time-change (Yt, Px)

^{to another process}

of

^{the same} type ^is

a linear

time-change

^{t -}

kt,

^{k > o.}

Proof. -

^{What we have to show}

is,

^{that if}

At

^{is a} continuous additive functional of

(Yt)

^with

Tt

^{as the}

right

continuous inverse and

(YTt, ^P~)

^is

multiplicatively invariant,

^then there exists k > 0 such that

At = kt

^{for all}

t ~. According

^to

[6], (Yt, P")

^{has a} weak dual with

respect

^{to the}

measure dx. We can now

show, by

^{the same} way ^as in Theorem

2.1,

that

At

^{has a Revuz measure}

which

gives

the assertion.

’

Proof of

Theorem 2.3. - Let

(Zt, QZ)

^{be a transient}

Levy

process ^on

( - oo, +00). Then,

^as remarked in the

proof

^of

Proposition,

^{the weak}

dual

(Zt, QZ),

which also is ^a

Levy

process, is transient. Now

(exp Zt, "),

^{x >}

0,

^is

multiplicatively

invariant and

has,

^{as shown in}

[6],

a weak dual

(expZt,

^with

respect

^{to the measure dx. It is}

easily

seen that

(Zt)

is transient iff

(exp Zt)

is transient and so,

according

^to

Lemma, (exp Zt)

^{cannot be}

time-changed

^{to another}

multiplicatively

^invari-

ant process with "nice

paths"

otherwise than

by

the linear time

change t - kt,

k > o. Thus one to one

correspondence

between the class of

Levy

Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques

SELF-SIMILAR MARKOV PROCESSES 587

processes and the class of

multiplicatively

^invariant processes with "nice

paths" gives

the assertion.

REFERENCES

[1] R. M. BLUMENTHAL and R. K. GETOOR, ^Markov Processes and Potential Theory, ^New York, ^Academic Press, ^1968.

[2] ^{R. K.} GETOOR, Transience and Recurrence of Markov Processes, Séminaire de Probabili- tés XIV, 1978/79, pp. 397-409.

[3] ^R. K. GETOOR and M. J. SHARPE, Naturality, Standardness and weak Duality ^for

Markov Processes, ^Z. Wahrscheinlichkeitstheorie verw. Geb., Vol. 67, 1984, p. 1-62.

[4] S. E. GRAVERSEN and J. VUOLLE-APIALA, 03B1-Self-Similar Markov Processes, Prob. Th.

Rel. Fields, ^Vol. 71, 1986, pp. 149-158.

[5] ^{J. W.} LAMPERTI, Semi-Stable Markov Processes I, ^Z. Wahrscheinlichkeitstheorie verw.

Geb., Vol. 22, 1972, pp. 205-225.

[6] J. VUOLLE-APIALA and S. E. GRAVERSEN, Duality Theory ^for Self-Similar Processes, Ann. Inst. Henri Poincaré, Probabilités et Statistiques, ^Vol. 22, (3), 1986, pp. 323-332.

(Manuscript received November 22th, 1988) (accepted ^June 15th, 1989.)